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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 464814.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
464814.dz1 | 464814dz2 | \([1, -1, 1, -374954, -88278235]\) | \(23314974828101839/322524\) | \(80646158628\) | \([2]\) | \(2621440\) | \(1.6480\) | \(\Gamma_0(N)\)-optimal* |
464814.dz2 | 464814dz1 | \([1, -1, 1, -23414, -1377547]\) | \(-5676903560719/21172752\) | \(-5294183119344\) | \([2]\) | \(1310720\) | \(1.3014\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 464814.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 464814.dz do not have complex multiplication.Modular form 464814.2.a.dz
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.